The drift method utilizes the relationship that the expected steady-state drift is zero for well-behaved Lyapunov functions, i.e., \(\mathbb {E}\left[\Delta g\left({U}^{(n)}(\infty )\right)\right]=0\) .
One key to the drift method is to craft an appropriate Lyapunov function \(g\)  such that its drift, \(\Delta g\left({U}^{(n)}(\infty )\right)\) , decomposes into terms that correspond to the metric of interest (e.g., the total number of jobs in the system) and terms that are tractable to bound. There are mainly two general approaches to constructing such Lyapunov functions. The traditional approach [1]}, [2]}, [3]} is based on establishing state-space collapse, a property whereby the state is concentrated around a strict subset of the entire state space in heavy load. Another approach [4]}, [5]}, [6]}, [7]}, [8]} is based on solving the so-called Stein's equation after coupling the system with a simple fluid model.
